Summary

In Fig. 5.8, theG⁰ band intensities for combination of the iTO and LO phonon modes for all (n, m) SWNTs in the diameter range 0.6< d_t <1.6 nm have small values compared with those for the iTO+iTO phonon combination, even though the iTO+LO intensity gives the dependence of metallicity in the excitation laser energy range, 1.2 < E <1.75 eV. Therefore, we can say that the iTO+iTO combination for the G⁰ band is dominant compared with the iTO+LO combination.

Chapter 6

G ⁰ band of multi-layer graphene

In this Chapter, we present the electronic energy band structure for multi-layer graphene and calculate the Raman G⁰ band as a function of the number of graphene layers, and compare the calculated results with the experiment. As explained in Section 3.5, the inter-layer interaction in multi-layer graphene gives the energy band splitting near the Fermi level. Since the G⁰ band is a double resonance Raman scattering process, the split energy band structure for multi-layer graphene results in different sub-peaks in the G⁰ band region. Therefore, in order to understand the electronic energy band structure for multi-layer graphene, we need to analyze sub-peaks of theG⁰band in experimental Raman spectra.

6.1 Raman scattering processes

In the case of single layer graphene, we already described in detail the double resonance Raman scattering process in section 3.5. Since the energy dispersion for π electron of single layer graphene gives only two linear bands at the K point in the Brillouin zone of a graphene sheet, we just consider one possible optical scattering process. As the result, the G⁰ band shows a sharp peak with small spectral width.

In double layer graphene, the energy band around the K point shows two parabolic bands for up and down bands which are split from two linear band of single layer graphene due to inter-layer interaction between π electrons (see Fig. 3.4 (a)). The splitting of the energy band around the K point plays an important role to determine the G⁰ band as explained above. For the G⁰ band calculation of double layer graphene, we have to find

95

K

2.41 eV φ P

^λ

c

₂

c

₁

v

₁

v

₂

<v₁H

el-opc₁>

<v₂H

el-opc

2>

<v₂H

el-opc₁>

<v₁H

el-opc₂>

××

K

2.41 eV φ P

^λ

c

₂

c

₁

v

₁

v

₂

<v₁H

el-opc₁>

<v₂H

el-opc

2>

<v₂H

el-opc₁>

<v₁H

el-opc₂>

××

Figure 6.1: Possible optical absorption processes (left top). By the optical matrix elements calculation, the optical absorption processes from v_i to c_i (i = 1,2) are possible, where i indicates the energy sub-band index. The light polarization vector P^λ is in the zigzag direction (left bottom) and the optical matrix elements is given as a function of polar angle φalong the contour ofE_L = 2.41 eV (right). Black, red, blue, and green arrows correspond to the electron-photon matrix elements ­

v₁|H_el−op|c₁® , ­

v₂|H_el−op|c₂® , ­

v₂|H_el−op|c₁® , and

­v₁|H_el−op|c₂®

, respectively [72].

possible Raman scattering processes. First, we check possible electron-photon absorp-tion processes, when a valence electron is excited to the conducabsorp-tion band of double layer graphene, as shown in Fig. 6.1. In order to calculate the electron-photon matrix ele-ment M_el−op for double layer graphene, we have to consider four optical transitions from the valence band to the conduction band. For double layer graphene, the optical matrix elements M_el−op(v₁ to c₂) andM_el−op(v₂ to c₁) become almost zero by theoretical calcu-lation which can be understood by the symmetry, as shown in right plot of Fig. 6.1. The electron-photon matrix elements calculation is based on Section 3.1. In comparison, the optical matrix elements M_el−op(v₁ to c₁) and M_el−op(v₂ to c₂) within the same layer are similar to that for single layer graphene. Thus, we have two possible optical absorption

c

2

c

1

v

1

v

2

E_L

2E_P11

P

₁₁

k₁ k'₁

c

2

c

1

v

1

v

2

E_L

2E_P12

P

₁₂

k₁ k'₂

c

2

c

₁

v

₁

v

₂ E_L

2E_P22

P

₂₂

k₂ k'₂

K M K'

Γ Γ

c

2

c

₁

v

1

v

2

E_L

2E_P21

P

₂₁

k₂ k'₂

K M K'

Γ Γ

c

2

c

1

v

1

v

2

E_L

2E_P11

P

₁₁

c

2

c

1

v

1

v

2

E_L

2E_P11

P

₁₁

k₁ k'₁

c

2

c

1

v

1

v

2

E_L

2E_P12

P

₁₂

k₁ k'₂

c

2

c

₁

v

₁

v

₂ E_L

2E_P22

P

₂₂

k₂ k'₂

K M K'

Γ Γ

c

2

c

₁

v

1

v

2

E_L

2E_P21

P

₂₁

k₂ k'₂

K M K'

Γ Γ

q₁₁ q₁₂

q₂₂ q₂₁

Figure 6.2: Schematic double resonance Raman scattering processes of double layer graphene. We find four possible optical processes in the energy band structure of double layer graphene.

processes and we can make two equi-energy contour around the K point as explained in Section 3.5. An electron which is excited from v_i toc_i enery band scatters to one of two parabolic bands at the K⁰ point, emitting a phonon, where i(=1,2) indicates the energy sub-band index.

Therefore, we have four possible double resonance Raman scattering processes for the double layer graphene along the Γ−K−M−K⁰−Γ direction, as shown in Fig. 6.2. These four possible inter-valley double resonance Raman scattering processes might lead to the observation of four sub-peaks in the double layer graphene. P₁₁process represents that an electron with wave vector k1 in v1 band scatters to c1 band, absorbing the incident laser energy E_L, and scatters to another electronic state k⁰₁ in c⁰₁ band by emitting a phonon with wave vector q₁₁, and scatters back to c₁ band, and finally recombines with a hole in v1 band, where q11 =k1−k⁰₁. P22process involves the optical absorption process fromv2

E

_L

P

₁₂

k₁ k'₂

q

₁₂

E

_L

P

₁₂

k₁ k'₂

q

₁₂

E

_L

P

₁₃

k1 k'₃

q

₁₃

E

_L

P

₁₃

k1 k'₃

q

₁₃

E

_L

P

₂₁

k2

k'₁

q

₂₁

E

_L

P

₂₁

k2

k'₁

q

₂₁

E

_L

P

₂₂

k2

k'₂

q

₂₂

E

_L

P

₂₂

k2

k'₂

q

₂₂

E

_L

P

₂₃

k₂ k'

3

q

₂₃

E

_L

P

₂₃

k₂ k'

3

q

₂₃

K M K'

Γ Γ

E

_L

P

₃₁

k₃

k'1

q

₃₁

K M K'

Γ Γ

E

_L

P

₃₁

k₃

k'1

q

₃₁

E

_L

P

₃₂

k₃ k'₂

q

₃₂

E

_L

P

₃₂

k₃ k'₂

q

₃₂

K M K'

Γ Γ

E

_L

P

₃₃

k₃ k'

2

q

₃₃

K M K'

Γ Γ

E

_L

P

₃₃

k₃ k'

2

q

₃₃

K M K'

Γ Γ

c3

c₁

v₁ v₃

E

_L

P

₁₁

c₂

v₂

k₁ k'₁

q

₁₁

c3

c₁

v₁ v₃

E

_L

P

₁₁

c₂

v₂

k₁ k'₁

q

₁₁

2E_ph

Figure 6.3: Schematic double resonance Raman scattering processes of triple layer graphene. We find nine possible optical processes in the electronic band structure of triple layer graphene.

to c₂ band with electron wave vector k₂ and the electron-phonon scattering process from c₂ toc⁰₂ band by emitting a phonon with wave vectorq₂₂=k₂−k⁰₂. Similarly, P₁₂and P₂₁ processes also involve the optical process of an electron with k₁ and k₂, respectively, and involve the electron-phonon scattering process for a phonon with wave vector q₁₂ andq₂₁, respectively, where q₁₂ =k₁−k⁰₂, and q₂₁=k₂−k⁰₁. As explained in Chapter 5, the G⁰ band comes from the overtone of iTO phonon mode. Since the wave vector of iTO phonon mode q₁₁ associated with P₁₁ process is the largest of these four wave vetors, as shown in Fig. 6.2, the P₁₁ process corresponds to the highest frequency subpeak in theG⁰ band of double layer graphene. On the other hand, the P22 process gives the lowest frequency

2600 2700 2800

Raman shift (cm

^-1

)

Intensity (arb. units)

2600 2700 2800

Raman shift (cm

^-1

)

iTO+iTO Exp.

1.65 1.75

1.92 2.06

2.41

1.83 1.92

2.06 2.41

Figure 6.4: Theoretical (left) and experimental (right) G⁰ band spectra for single layer graphene for the different excitation laser energies. The number for each peak indicates the excitation laser energy. The experimental result is given by Mr. A. Reina in MIT in the collaboration with our group [26].

sub-peak due to the smallest phonon wave vector q₂₂. The inter-mediate sub-peaks are associated with the P₁₂and P₂₁ processes. The ETB calculation for the energy dispersion relations for the double layer graphene gives same phonon wave vector for q₁₂ and q₂₁. Thus, the inter-mediate sub-peaks for the P₁₂ and P₂₁ processes are degenerated at same frequency.

For the case of triple layer graphene, we can also neglect the optical matrix elements which connect different subband indexes of the valence and conduction bands, because of the appearance of a similar dipole selection rule to the double layer case. As the result, there are nine possible double resonance Raman scattering processes in triple layer graphene, as shown in Fig. 6.3. The P₁₁ and P₃₃ processes give the highest and lowest frequency sub-peaks in the G⁰ band spectra with corresponding the phonon wave vectors q11 and q33, where q11 = k1 −k⁰₁, and q33 = k3 −k⁰₃. The other sub-peaks in the G⁰ band spectra for the triple layer graphene are associated with the degenerated processes P₂₁+ P₁₂, P₂₂+ P₁₃+ P₃₁, and P₂₃+ P₃₂, which give similar phonon wave vectors to one another for each sub-peak.

2600 2700 2800

Raman Shift (cm

^-1

)

Intensity (a.u.)

1.65

1.75 1.92

2.41 2.06

Figure 6.5: The G⁰ band calculation for excitation laser energies E_L =1.65, 1.75, 1.92, 2.06, and 2.41 eV, for single layer graphene without considering E_L factor in the optical matrix elements of Eq. (3.1.4). Thus, the calcuated G⁰ band intensity as a function of the excitation laser energy becomes similar to the experiment in right panel of Fig. 6.4.

ドキュメント内 Double resonance Raman spectroscopy of single wall carbon nanotubes and graphene (ページ 101-108)